Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $r \neq 0$. $x = \dfrac{-2}{r(r - 8)} \div \dfrac{9}{5r - 40} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{-2}{r(r - 8)} \times \dfrac{5r - 40}{9} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ -2 \times (5r - 40) } { r(r - 8) \times 9 } $ $ x = \dfrac {-2 \times 5(r - 8)} {9 \times r(r - 8)} $ $ x = \dfrac{-10(r - 8)}{9r(r - 8)} $ We can cancel the $r - 8$ so long as $r - 8 \neq 0$ Therefore $r \neq 8$ $x = \dfrac{-10 \cancel{(r - 8})}{9r \cancel{(r - 8)}} = -\dfrac{10}{9r} $